Papers

We study the magnetic deformation of the $(2+1)$d Ising CFT, also known as Ising Field Theory (IFT), on a spatial sphere. We use the Fuzzy Sphere (FS) Ising setup and compare it to the Truncated Conformal Space Approach (TCSA). Universal, infinite volume, IFT quantities are extracted by modeling the effects of curvature, as expected from a local effective theory on the sphere. We find evidence that IFT contains a bound-state with a small binding energy. Locality also allows us to extract the same IFT quantities from different angular momentum sectors, providing additional consistency checks.

We identify the precise F-theory backgrounds dual to the heterotic string theory compactified on $T^2$ with general Wilson lines turned on for one of the two $E_8$'s. The elliptic curve describing the backgrounds is expressed in terms of orthogonal modular forms and is determined so that it admits a nontrivial scaling limit yielding a rational elliptic surface. We confirm that the curve reduces to the Seiberg-Witten curve for the E-string theory in this limit. We conjecture that our result gives an explicit inverse period map for algebraic K3 surfaces polarized by the $U\oplus E_8(-1)$ lattice.

Recent developments in the Swampland program motivate the existence of a mesoscopic Dark Dimension of size $0.1-10~\mu$m with bulk Majorana fermions in the $1-10$ keV range. Motivated by this, we derive terrestrial constraints on Majorana bulk neutrinos as a function of the compactification radius. After determining the mass spectrum and mixing structure, we confront the model with Daya Bay neutrino-oscillation data, the KATRIN beta-decay bound, and the KamLAND-Zen neutrinoless double beta decay limit. For the latter, we obtain a closed-form expression for the effective Majorana mass and show that the Kaluza-Klein tower efficiently screens neutrinoless double beta decay when the nuclear momentum exceeds the compactification and Majorana scales. In the Dark-Dimension window, beta decay provides the strongest constraint, pushing the allowed Yukawa couplings down to $\mathcal{O}(10^{-3})$. Oscillation and neutrinoless double beta decay searches remain complementary, probing smaller and larger Majorana masses, respectively. For completeness, outside the Dark-Dimension assumptions, we investigate how hierarchical Majorana masses can weaken the constraints.

We derive an entropy formula for recently proposed tensor network models which prepare diffeomorphism invariant states of topological quantum field theories with non-compact and/or continuous gauge groups. We show that our entropy formula generalizes the notion of “topological entanglement entropy” to incorporate the infinite number of particle-like excitations in such theories. When our networks are endowed with gauge group $\mathrm{SL}(2,\mathbb{R})$, we can interpret them as models of three-dimensional gravity with small Newton's constant and possibly non-invertible metrics.

We study metric-source responses of celestial Einstein amplitudes while retaining their momentum-conservation distribution. The consecutive opposite-helicity response is related to the simultaneous gravitational contact at arbitrary hard multiplicity. Their difference contains a Lorentz transformation that annihilates the conservation delta function, determining the normal recoil contribution and its relation to the undifferentiated coefficient. A six-graviton tree calculation evaluates both coefficients on a real scattering channel. For the summed homogeneous contact, momentum conservation cancels the leading angular singularities. Uniform bounds at coincident soft directions and hard punctures yield integrated endpoint estimates that determine the ordered Mellin residues after smooth source smearing. The mixed Mellin divisor also retains logarithms of soft–hard invariant ratios at generic conformal weight. In Weyl–Beltrami variables, the quadratic reference Ward response combines soft-leg, shadow-map and spin-frame terms into the transported metric Hessian, with a nonzero symmetric mixed susceptibility. Relative shadow primitives require boundary currents when represented across logarithmic cuts; their source variation fixes the accompanying contour contribution. Joint factorization and angular expansions organize the additional NMHV relative-contour terms into residue coefficients that distinguish metric integrability from matching to a specified source functional. An independent vacuum completion supplies the Euler response, which fixed-area harmonic combinations separate from local curvature terms at bounded derivative order.

Violations of Bell's inequalities offer a definitive signal of non-classical (quantum) behavior in local deterministic systems. Yet, in many physical settings where quantum mechanics is expected to play an important role, one cannot construct a Bell-type test using the available observables. Cosmology offers one concrete example, where cosmic structure may have originated from quantum vacuum fluctuations yet all observations are effectively classical. Nevertheless, recent work suggests that quantum and classical time evolution may still be differentiated by the pattern of statistical correlations of these observables. In this paper, we explore and generalize the use of correlations as Bell-type test of the nature of time evolution. We show that quantum vacuum fluctuations of closed systems undergoing Hamiltonian evolution produce unique correlations that are not mimicked by classical Hamiltonian systems. Correlations in the quantum vacuum are generally time-independent and are controlled by the energy gap between the ground and exited states. Classical systems exhibit apparent poles at physical frequencies that do not arise in the quantum vacuum. As one approaches the poles, the evolution becomes dominated by a resonant Hamiltonian giving rise to time dependent correlations that eventually decay through dephasing. Finally, we show that this behavior is distinct from other signals of quantum evolution, including applications to quantum optics, quantum walks, and quantum search.

Point particles are generically ill defined in fully nonlinear general relativity, although they are naturally well defined in linear perturbation theory. In a previous paper, two of us showed how they can be rigorously defined from matched asymptotic expansions at second order in perturbation theory on a generic background spacetime. Here we show how the resulting field equations can be recast in a more practical, more fully “skeletonized” form. We also show that this form of the field equations enables simple derivations of equations of motion. Finally, we show how calculations in second-order perturbation theory can bypass the use of singular “punctures” and make rigorous use of off-the-shelf regularization methods such as Blanchet and Damour's Hadamard regularization, opening new avenues to second-order self-force calculations in black hole binaries.

Reciprocal contacts can favor cooperation by protecting a lineage until its descendants compete as a group. An individual is chosen to reproduce from the whole population, with a probability increasing with its averaged cooperation payoff. Its offspring replaces a neighbor through contacts weighted equally in both directions. We compare a single cooperator taking over defectors with the reverse invasion, using the same distribution of introduction sites. When payoffs weakly affect reproduction, every network with at least six sites and a path visiting them all admits weights favoring cooperation at any benefit-cost ratio $b/c>1$. This holds for mutants placed uniformly or arising in newborn offspring. Unequal contacts sustain aligned domains around two protected sites, converting individual reproduction into collective competition. Cooperation cannot have an advantage when its benefit does not exceed its cost, at any positive selection strength with positive reproduction rates. Where mutants arise changes which contact patterns can use this mechanism. On a five-site path, the weak-selection threshold has the sharp lower bound seven under uniform placement; newborn introduction never favors cooperation at this order. We classify the contact patterns with at most five sites that can favor cooperation under either rule; newborn introduction requires a four- or five-site cycle. Tracing copies backward to their ancestors separates domain competition from the probability of reaching it. Exact polynomial inequalities prove sharp threshold bounds for paths and the four- and five-site cycles. The same ancestry reconstructs evolution without selection on weighted paths and cycles and fixes the slowest relaxation rate on every connected reciprocal network. Cooperation thus depends on both the collective competitors created by contacts and the route by which a new lineage enters them.

The spatial distribution of anomalies in extra dimensional theories on a slice of AdS leads to a puzzle where anomalies may not match across different geometries that are dual to various states of the same CFT. We argue that consistency of the anomaly across such geometries necessitates the addition of topological terms or other bulk fermions to the theory that flow any IR localized consistent anomaly into the “UV" region of these geometries if one wishes to have such a dual interpretation.

We prove that the minimal null string theory on a background manifold ${\mathcal M}$ is classically equivalent to the ILST theory on the projective boundary of $\mathcal M$, ${{\cal B}_{_{\text{P}}}}={\partial_{_{\text{P}}}\mathcal{M}}$. We conjecture that this equivalence also holds at quantum level, we dub this correspondence null string holography. In particular, we show the minimal null string theory on the Poincaré patch of a $D$ dimensional Anti de Sitter (AdS$_D$) background is described by the ILST theory on $(D-1)$-dimensional Minkowski background, paralleling the celebrated AdS/CFT. The minimal null string theory on the cosmological patch of dS$_D$ is governed by the ILST theory on $(D-1)$-dimensional flat Euclidean space. For null strings on a $D$-dimensional Minkowski space $\mathbb{M}_D$, depending on the coordinate patches used to span $\mathbb{M}_D$, the projective boundary includes a dS$_{D-1}$, asymptotic null infinities ${\mathcal I}^\pm$ and the hyperboloid $\mathbb{H}^{D-1}$ patches. The ILST theory dual to the minimal null strings on each patch, then resides on the piece of the projective boundary spanned by the coordinates adopted in the patch. We comment on possible applications of null string holography to black hole horizons.

We provide a unified analysis of scalar-field quantization and detector response in relativistic rotating frames, comparing rigid and Trocheris-Takeno (TT) descriptions in unbounded Minkowski spacetime. We distinguish three questions that are often conflated: (i) how a fixed quantum state is represented in rotating coordinates, (ii) whether rotating time evolution selects a global ground state, and (iii) how a rotating detector responds to the vacuum. Coordinate-transformed rigid and TT modes span the same positive-frequency subspace as inertial modes: their Bogoliubov beta coefficients vanish and their mode sums reproduce the Minkowski Wightman function. Neither rotating time flow selects a global scalar ground state, although for different reasons: the rigid generator is not globally timelike and is unbounded below, whereas TT time translation is not a Killing symmetry. Nevertheless, a circular Unruh-DeWitt detector in the Minkowski vacuum has a stationary, nonzero response. We relate this response to negative corotating frequencies, evaluate it after an explicit Hadamard subtraction, and show that it is nonthermal. Thus vanishing Bogoliubov mixing, the absence of a symmetry-selected rotating vacuum, and detector excitation are mutually consistent.

This work concerns a procedure for removing gauge dependence from graviton corrections to the effective field equations for a massless, minimally coupled scalar on de Sitter background. The procedure involves combining diagrams in the same way as the scattering amplitude for the $t$-channel exchange of the massless scalar between two massive particles, but without taking the asymptotic limits which are problematic in cosmology. Implementing this at 1-loop on flat space background requires five classes of diagrams, in addition to the naive exchange, and the combination of those diagrams does eliminate dependence on the graviton gauge fixing functional. When those same diagrams are generalized to de Sitter background, their combination cancels the most important (“nonlocal”) secular gauge dependence, but it leaves (“local”) secular gauge dependence that is associated with external legs. Local secular gauge dependence can be removed by adding two additional classes of diagrams, and making secular corrections to the external wavefunctions. The purpose of this paper is to include these additional corrections on de Sitter background in order to derive fully gauge independent effective field equations at 1-loop order.

We develop an on-shell description of low-frequency gravitational waveforms from black-hole mergers beyond leading order. Treating the strongly coupled merger as effective hard $S$-matrix data, we organize its long-wavelength response using soft theorems and the KMOC formalism. At next-to-leading order, the quantum soft theorem contains logarithmic terms absent from the classical soft theorem. We show that these extra terms cancel in the full KMOC in-in observable between the one-loop radiative amplitude and the corresponding graviton cut, leaving precisely the classical logarithmic contributions associated with gravitational drag and early-time acceleration. We also identify the $1/\omega$ corrections from remnant recoil and Christodoulou non-linear memory. These results reveal a hierarchy of merger information accessible at low frequency: logarithmic tails depend only on asymptotic hard data, recoil probes total radiated momentum, while non-linear memory probes the angular distribution of the emitted radiation.

For the first time, we establish a direct connection between inflationary dynamics and the subsequent magnetohydrodynamic evolution. Starting from vacuum fluctuations, we self-consistently evolve the coupled inflaton-plasma system through inflation, reheating, and into the radiation-dominated era. We find an inflaton-driven inverse cascade followed by freely decaying turbulence, with kinetic energy dominating over magnetic energy and driving dynamo amplification. The inflaton-driven turbulence causes steeper magnetic energy decay than helical turbulence and should therefore be accounted for subsequent evolution of primordial fields and their present-day observables.

We construct a holographic dark-energy model using Kaniadakis entropy with the particle horizon as the infrared cutoff, consistently modifying both the holographic density and the Friedmann background. In standard Einstein gravity, noninteracting particle-horizon holographic dark energy (HDE) does not produce the sufficiently negative pressure required for late-time accelerated expansion. We show that the Kaniadakis deformation changes this behavior while the particle horizon is retained. For the representative parameter choice $K=1.90\times10^{-36}$, with $\Omega_{\rm DE0}=0.7$ and $c^2=0.64$, the deceleration parameter changes sign at $z\simeq0.585$ in the noninteracting case. Including the interaction strengths $b^2=0.03$ and $0.06$ changes the transition only slightly, giving $z\simeq0.586$ and $0.589$, respectively. Thus, for the parameter set considered here, the interaction does not generate the acceleration; rather, the transition is already present in the noninteracting Kaniadakis model and the interaction produces only a small shift in its timing. We derive the autonomous evolution equations, the effective dark-energy equation of state, and the deceleration parameter. The adiabatic squared sound speed is also examined as a diagnostic of the effective-fluid stability, while statefinder variables are used to characterize deviations from $\Lambda$CDM. Finally, we verify that the $K\to0$ limit continuously recovers standard Einstein-gravity particle-horizon HDE, for which the noninteracting model remains decelerating.

A critical component of relativistic three-body scattering amplitudes is the one-particle exchange (OPE) process, wherein a single particle is exchanged between incoming and outgoing two-body subsystems. We present an improved, finite-sum expression for the partial-wave OPE that is valid for three massive, spinless particles in arbitrary angular momentum configurations. A collection of analytic and numerical checks demonstrate that our finite-sum expression reproduces the known properties of the partial-wave OPE, including its threshold scaling behavior and singularity structure. Compared with previous work, this result more efficiently handles the rapid proliferation of partial waves contributing to any total angular momentum $J$, enabling the construction of robust wavesets for three-body amplitude analyses.

We discuss orbifold compactifications of the type II string, where the orbifold combines a free action on a base manifold and the $(-1)^{\boldsymbol{F}_{\text{L}}}$ symmetry which acts by -1 on left-moving Ramond sectors. Such backgrounds can preserve minimal supersymmetry and lift to M-theory compactification on smooth unorientable manifolds.

Our central premise is that quantum mechanics may be the statistical limit of a more fundamental discrete theory: any such theory equips a physical system with a finite information capacity, and its departure from quantum statistics is controlled by how much of that capacity the system uses. We show that commercial cloud photonic quantum processors have reached the precision required to bound this capacity from below, using the binary-sequence model of Powers et al. as the concrete test theory: outcome probabilities arise from counting discrete sequences of length $n$, quantum mechanics is recovered as $n \to \infty$, and $n$ measures the information capacity of the register behind a prepared state. Photon Fock states $|1\rangle$, $|1,1\rangle$, heralded $|2\rangle$, and cascaded beam-splitter pairs are measured on programmable interferometers with dominant systematics determined in situ. The model's composition-consistent parametrization, singled out by requiring that rotations compose, recovers quantum mechanics with deviations $1.24/n$; a random-effects likelihood analysis calibrated by parametric bootstrap excludes all $n \le 100$: the information capacity of the register carrying the two-photon state, if finite, exceeds $10^2$. Cascaded beam splitters test the composition law directly: the data are split-invariant, excluding naive count composition at $8\sigma$ and confirming the interference-sign rule. Model-independently, curve-averaged deviations from the quantum partition law larger than $2.3\times10^{-2}$ are excluded at 95% CL, and the originally published linear parametrization is excluded outright. Because the compilation offset is frozen per circuit it is calibratable, opening the $10^{-3}$ floor ($n \sim 10^3$) to current hardware: cloud photonic processors are quantitative instruments for quantum foundations, and information capacity an experimentally boundable quantity.

In the late 1990s, Suzuki and Yasuta proposed a compact formula that extracts the decay rate per unit volume of a false vacuum in the $D$-dimensional $O(N)$ $\lambda(\phi^2)^2$ theory with an unbounded potential from conventional perturbative coefficients of the vacuum energy density, i.e., vacuum bubble diagrams. The idea was to identify the imaginary part arising from the Borel integral along the discontinuity of the Borel transform with that of the vacuum energy density. While the formula works quite well for~$D=1$, i.e., quantum mechanics, its validity remained unclear for~$D\geq2$ because only the first three perturbative coefficients for~$D=2$ were available and the result showed no sign of convergence. In the present paper, we reexamine this approach using the first seven nontrivial perturbative coefficients (up to nine loops) for~$D=2$ and~$N=1$ obtained by Serone, Spada, and~Villadoro. Introducing two tunable parameters in the finite-order truncated Borel transform following these authors, we find that the imaginary part converges as the perturbative order increases, with the last few orders agreeing to within a few percent. In the intermediate range of the coupling constant, $3\lesssim\widetilde{g}\lesssim8$, this approach yields an imaginary part rather close to the leading-order semi-classical approximation with the one-loop determinant; it is $10$–$20\%$ larger than the semi-classical result with the two-loop correction computed by Malatesta, Parisi, and~Rizzo.

The Low-Burnett-Kroll theorem (extended to non-abelian gauge theory) states that the tree-level emission amplitude of a soft gauge boson from an arbitrary hard process can be expressed in terms of the non-radiative amplitude including the next-to-soft term. Reformulating the soft theorem in the framework of an on-shell effective theory leads to a natural gauge-invariant split into an external and internal emission process to any order in the soft expansion. In this framework, we derive a compact expression for the all-order external emission amplitude expressed in terms of the non-radiative amplitude, and express the local emission amplitude through a single gauge-invariant operator at every order beyond the next-to-soft one. For scalar QCD, we compare the number of Lorentz- and $SU(3)$-invariant amplitudes for the radiative and non-radiative hard process with an arbitrary number of scalar quarks and antiquarks.

We investigate the impact of self-interactions on the Rényi and entanglement entropies of a scalar field on $(2+1)$-dimensional spacetimes, whose spatial sections are modeled by fuzzy spaces, specifically the fuzzy sphere and the fuzzy disc. We compute the first-order perturbative correction induced by a $\lambda\phi^4$ interaction using the Green's function approach. In contrast to the free theory, where the entanglement entropy is dominated by degrees of freedom near the entangling boundary and obeys an area law, we find that the interaction correction has an extensive bulk contribution, receiving significant contributions from degrees of freedom throughout the fuzzy space. For the fuzzy sphere, the correction exhibits strong infrared sensitivity associated with the zero mode. We isolate and resolve this zero-mode IR divergence by projecting out the zero mode, thereby obtaining a physically meaningful quantity. In the commutative continuum limit, the interaction correction has the same degree of UV divergence as the free entropy but does not obey a pure area law. Furthermore, we analyze the Moyal plane limit, where the interaction correction exhibits a distinct IR divergence. We discuss the physical origin of these extensive bulk features and examine their possible connection to the celebrated UV/IR mixing phenomenon in noncommutative quantum field theories.

We consider 4-dimensional asymptotically flat vacuum spacetimes near future null infinity endowed with the most general allowable Carroll geometry. We show that the near-boundary radial expansion (to a certain order) can be organised in terms of connections that can be obtained by gauging the conformal Carroll algebra. The only non-vanishing curvatures in this gauging procedure are those that are associated with the special conformal generators and we will refer to these as the $K$-curvatures. The vanishing of these $K$-curvatures defines an asymptotic vacuum spacetime and we use this to construct a boundary (i.e. Carroll) covariant expression for the vacuum (soft) shear in terms of two boundary Carroll scalar fields. The $K$-curvatures transform in a hierarchical fashion into one another under Carroll boosts. This leads to a classification of 4 types of spacetimes: vacuum, strongly and weakly non-radiative, and radiative spacetimes. It is shown that the $K$-curvatures correspond to 5 of the 10 Weyl tensor components at leading order in their $1/r$ expansion. We furthermore observe that one of the $K$-curvatures is equal to the recently found Carroll boost anomaly. Finally, we show that the Bondi loss equations for the boundary energy-momentum-news complex can be cast into a form involving another energy-momentum tensor with vanishing energy flux that is traceless and whose non-conservation is entirely captured by the $K$-curvatures. The BMS currents can be obtained by contracting this latter energy-momentum tensor with a Carroll conformal Killing vector.

Can propagating gravitational waves serve as operational probes of the quantum nature of gravity? We address this question by developing a unified theoretical framework that combines spacetime geometry, quantum information, and gravitational-wave physics. Starting from the geodesic deviation equation in linearized General Relativity, we derive the effective detector Hamiltonian directly from spacetime geometry and construct the complete quantum dynamics for detector subsystems interacting with both classical and quantized propagating gravitational-wave fields. This unified formulation enables a direct comparison between classical and quantum descriptions of gravitational radiation within the same physical framework. We demonstrate that classical gravitational-wave backgrounds can induce mixedness in the detector state but cannot generate genuine quantum correlations between the detector subsystems. In contrast, quantized gravitational waves coherently mediate gravity-induced entanglement, quantum coherence, quantum memory, and nonclassical correlations, providing clear operational signatures of the quantum nature of propagating gravitational radiation. We further discuss how mesoscopic quantum mechanical oscillators offer a promising route towards experimentally probing these effects. Our results establish a geometric and quantum-information-based framework for exploring quantum gravity through propagating gravitational waves.

We determine $2,540$ unknotting numbers of prime knots with at most $13$ crossings, that are unknown in the KnotInfo snapshot of 9 September 2026. The lower-bound calculations use Heegaard Floer correction-term obstructions and Greene's spanning-tree model. We also give explicit crossing-change constructions for upper bounds. For $959$ alternating knots with range $[2,3]$, we verify that no crossing change in a minimal diagram gives a knot of unknotting number one. We determine the unknotting numbers of all four knots in Brittenham and Hermiller's construction and thereby identify 13n3370 as an explicit counterexample to the original Bernhard-Jablan conjecture.

Under explicit hypotheses on the spectral Looijenga construction, we compute the hyperbolic-plane value of the zero-section invariant of Gukov, Krushkal, Meier, and Pei in periodic topological modular forms. The value is the Hopf element eta, and adjoining a hyperbolic plane acts by multiplication by this element. Earlier work obtains the hyperbolic value under an additional cobordism-duality assumption; here we derive it directly from the Looijenga restriction maps without that assumption. Together with annihilation and vanishing results for definite lattices, this gives a complete value formula on smooth closed simply connected spin four-manifolds. In particular, nonzero signature forces the invariant to vanish, so both orientations of a K3 surface have value zero.

We investigate whether quantum fluctuations of a circular probe string develop a quantum-chaotic response while traversing a wormhole throat. The classical circular-string embedding is periodic and radially stable, but its two physical transverse polarizations experience time-dependent tidal potentials. Expanding the world-sheet action to quadratic order, we canonically quantize these modes and construct out-of-time-ordered correlator(OTOC) amplitudes from their unequal-time commutators. For the Ellis–Bronnikov wormhole, both polarizations exhibit finite intervals of approximately exponential OTOC growth associated with the first throat passage. The corresponding dimensionless rate measured with respect to physical time is positive over the parameter range studied and generally decreases as the probe energy is increased relative to the throat scale. In the global-monopole extension, increasing the solid-angle deficit narrows the band of locally amplifiable modes and suppresses the extracted rates; a sufficiently strong defect can nearly quench the radial signal, while the angular channel retains a polarization-dependent non-monotonic structure when the energy-to-throat-scale ratio is small. These quantities characterize finite-time dynamical sensitivity in the Gaussian fluctuation sector and should not be identified with asymptotic many-body chaos or a thermodynamic phase transition. Although the numerical analysis uses two representative wormhole geometries, the construction depends only on covariant world-sheet fluctuations and real-time commutators. It therefore provides a transferable, non-holographic framework for applying quantum-chaos diagnostics directly to quantum probes in curved spacetimes.

We study the dual quantum symmetry $\mathrm{Rep}(G)$ in the two-dimensional theory $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$ obtained by gauging a non-abelian symmetry $G$ of $\mathfrak{T}_G$, where for Lie groups $G$ the gauging is understood as flat gauging. We develop a general framework for computing partition functions of $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$ in arbitrary non-invertible symmetry backgrounds, represented by topological defect networks of $\mathrm{Rep}(G)$, in terms of the partition functions of the original theory $\mathfrak{T}_G$. We also derive the inverse transformation, expressing partition functions of $\mathfrak{T}_G$ in $G$ backgrounds in terms of those of $\widetilde{\mathfrak{T}}_{\mathrm{Rep}(G)}$. We test the framework in several examples, including finite groups with multiplicity-free and higher-multiplicity fusion rules, and discuss a formal extension to compact Lie groups, focusing on $\mathrm{Rep}(SU(2))$.

We show that the singularities which the analytically continued double-well instanton develops in real (Minkowski) time are not an inherent feature of real-time tunneling solutions. Rather, they can be viewed as arising from a singular limit of a one-parameter family of regular real-time solutions. Using a linear homotopy between a triple-well and a double-well potential, we construct a family of potentials, parametrized by $p\in(0,1)$, whose real-time ($\alpha=0$) instanton solutions are everywhere regular and bounded. For any Wick rotation angle $\alpha\in(0,\pi/2)$, we prove that regularity is generic up to a measure-zero set. There exists a countably infinite, closed-form family of potential parameters $\{p^{\mathrm{singular}}_{k}(\alpha)\}_{k\geq0}$ at which the instanton becomes singular at exactly two points of the rotated time axis, never more. In the strict double-well limit, $p\to 1$ taken at $\alpha=0$, the entire family collapses onto the single point $(p,\alpha) = (1,0)$ at which the singular pairs of all $k$ occur simultaneously. This phenomenon produces the infinite singular comb found by Cherman and Ünsal. By solving the complexified equations of motion in terms of Weierstrass elliptic functions, we trace this comb to classical imaginary turning points receding to $\pm\mathrm{i}\infty$, giving a unified geometric and algebraic account of when, and exactly how often, real-time instantons become singular. We further show that the action of the continued instanton is independent of $\alpha$ only in its real part: its imaginary part jumps by one real-time period action at each singular angle, the $\alpha$-independence found by Cherman and Ünsal being the degenerate case of the double well, in which no such jump occurs.

We show that a ringdown waveform can admit a constant-coefficient quasinormal-mode (QNM) representation while its individual QNM-pole contributions remain driven. To establish this result, we construct a first-principles rational approximation to the Kerr Green's function from its QNM poles, physical residues, and horizon-frequency zeros. For particles plunging from the innermost stable circular orbit into Kerr black holes with spins 0.5-0.9, each QNM-pole contribution initially follows the source's instantaneous complex frequency and decouples only after the source decays faster than the corresponding mode. At late times, the source approaches a sum of damped oscillations at the third and higher horizon frequencies. The corresponding zeros in the Green's function cancel these source-frequency components in the coherent waveform, while the continuing drive generates oscillations at QNM frequencies. Thus, an apparently free QNM superposition can emerge before its individual contributions dynamically decouple from the source.

Starting from the known differential expansions, we express the HOMFLY-PT polynomials of twist and antiparallel double-braid knots in rectangular representations \([r^s]\) as determinants of \(r\times r\) and \(s\times s\) matrices. We first give an elementary derivation for the figure-eight knot and then re-sum the KNTZ formulas for twist and double-braid evolution. The resulting matrices are constructed from single-hook evolution coefficients and explicit representation-dependent weights. These formulas provide a starting point for investigating possible extensions of knot polynomials to KP/Toda \(\tau\)-functions. Extensions to non-rectangular representations and to more general knot families remain open problems.